RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA MARIA SILVIA LUCIDO Prime graph components of finite almost simple groups Rendiconti del Seminario Matematico della Università di Padova, tome 102 (1999), p. 1-22 <http://www.numdam.org/item?id=RSMUP_1999__102__1_0> © Rendiconti del Seminario Matematico della Università di Padova, 1999, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Prime Graph Components of Finite Almost Simple Groups. MARIA SILVIA LUCIDO (*) ABSTRACT - In this paper we describe the almost simple groups G such that the prime graph T( G ) is not connected. We construct the prime graph T( G ) of a fi- nite group G as follows: its vertices are the primes dividing the order of G and two vertices p, q are joined by an edge, and we write p - q, if there is an ele- ment in G of order pq. 1. Prime graph of almost simple groups. If G is a finite group, we define its prime graph, flG), as follows: its ver- tices are the primes dividing the order of G and two vertices p, q are joined by an edge, and we write p °--° q, if there is an element in G of order pq. We denote the set of all the connected components of the graph T( G ) = if of we de- by 1,,r i (G), for i 1, 2 , ... , ~(G)} and, the order G is even, note the component containing 2 by ,~ 1 ( G ) _ ,~ 1. We also denote by .7r(n) the set of all primes dividing n, if n is a natural number, and by :r(G) the set of vertices of r(G). The concept of prime graph arose during the investigation of certain cohomological questions associated with integral representations of fi- nite groups. It turned out that T( G ) is not connected if and only if the augmentation ideal of G is decomposable as a module. (see [4]). In addi- tion, nonconnectedness of T( G ) has relations also with the existence of isolated subgroups of G. A proper subgroup H of G is isolated if H n (*) Indirizzo dell’A.: Dipartimento di Matematica Pura e Applicata, Universi- ta di Padova, via Belzoni 7, 1-35131 Padova, Italy. E-mail: [email protected] 2 f1 H9 = 1 or H for every g E G and CG (h) ~ H for all h E H. It was proved in [10] that G has a nilpotent isolated Hall ;r-subgroup whenever G is non- soluble and R = Ri(G), i > 1. We have in fact the following equiva- lences : THEOREM [8]. If G is a finite group, then the following are equivalent: (1 ) the augmentation ideal of G decomposes as a module, (2) the group G contains an isolated subgroup, (3) the prime graph of G has more than one component. It is therefore interesting to know when the prime graph of a group G is not connected, i.e. has more than one component. The first classifi- cation is a result of Gruenberg and Kegel. THEOREM A [8]. If G is a finite group whose prime graph has more than one components, then G has one of the following structures: (a) Frobenius or 2-Frobenius; (b) simple; (c) an extension of a ~c 1-grou~ by a simple group; (d) sirnpLe by ,~ 1; (e) Jr i by simply by n 1. The case of solvable groups has been completely determined by Gru- enberg and Kegel: COROLLARY [8]. If G is solvable with more than one prime graph components, then G is either Frobenius or 2-Frobenius and G has exact- ly two components, one of which consists of the primes dividing the lower Frobenius complement. Also the case (b) of a simple group has been described by Williams in [10], by Kondratev in [8] and by Iiyori and Yamaki in [7]. A complete list of the sim- ple groups with more than one component can also be found in [9]. In this paper we determine the case (d). The case (d) is in fact the case of an almost simple group with more than one component. A group G is almost sample if there exists a finite simple non abelian group S such that S ~ G ~ Aut (S). First we observe that if r( G) is not connected then also T(,S) is not connected. Then, using the Classification of Finite 3 Simple Groups and the results of Williams and Kondratev, respectively in [10] and [8], we consider the various cases. The main results are then Lemma 2 and Theorem 3, concerning spo- radic and alternating groups, and Theorem 5, concerning finite simple groups of Lie type. In particular we observe that almost simple groups which are not simple have at most 4 components (see Table IV). REMARK 1. If F(G) is not connected, and G has a non-nilpotent normaL subgroup N, then r(N) is not connected. PROOF. We suppose that T(N) is connected. As T( G ) is not connect- ed, there must be p in ;r(G) such that p ~ q for any q in ,~(N). Let P be a p-Sylow subgroup of G. If we consider K = NP, where P acts on N by conjugation, then K is a Frobenius group with kernel N, that must be nilpotent, against our hypothesis. In order to describe almost simple groups with non-connected prime graph it is therefore enough to consider groups G such that G ~ Aut (S) and S is a simple group with non-connected prime graph. A complete de- scription of the simple groups with non-connected prime graph can be found in [10] and [8]. We suppose S G. We use the Classification of Finite Simple Groups. Before beginning with a general study we want to treat a particular case. LEMMA 2. If S = As , there are four groups G1= ,S6 , G2 , G3 , G4 = = Aut (A6 ) such that S Gi ~ Aut (S). Then PROOF. Since Aut (S) /Inn (S) is isomorphic to the Klein group with 4 elements, there are three almost simple groups over S of order 2 [ S [. Let G1, G2, G3 be such groups. Then G1 S6 and 2 --- 3 in 7r(S6). Let 8 be an outer automorphism of S6 of order 2. Since Aut (A6 ) = Aut (Ss ), we can consider G2 , the subgroup of Aut (A6 ) generated by Inn (A6) and 0. Then G2 is a splitting extension of Inn (A6 ). Moreover 0 centralizes an el- ement of order 5 of A6 , and 0 can not centralize any element of order 3 of A6 because 0 exchange the two conjugacy classes of elements of order 3 of A6. Therefore 2 --- 5 in T(G2 ). 4 Let G3 be the other subgroup of Aut (As ) of order 2 ~ A6 ~ . Then G3 is a non-split extension of A6 and there is not any involution outside Inn (As ). Therefore its prime graph is the same as that of As. Let G4 be Aut (As ), then T(G4 ) is connected. From these observations we can deduce the structure of the prime graph of the groups Gi, i = 1, 2, 3, 4.. THEOREM 3. If G = Aut (S) with S an alternating group or a spo- radic then is. not connected if and only if G is one of the following groups and the connected components are as follows: PROOF. If = An, the alternating group on n letters, n # 6, we know that G = Sn . From [10] we observe that r(S) is not connected if and only if n = p , (p + 1 ), (p + 2) for some prime p. If n = p + 2 the element + (1, 2, ... , p) commutes with the element (p + 1, ~ 2) in Sn and then ~ro --- 2 in On the other hand if n = p, p + 1 the centralizers of all the elements of order p are exactly the cyclic subgroups of order p that they generate. If ,S is a sporadic group, then I GIS I = 2 and the result easily follows from the Atlas [2]. We now suppose that is a finite simple group of Lie type over the field with elements. We recall that, in this case, the connected components for i > 1 of are exactly sets 5 of type for some maximal torus T such that T is isolated (see Lemma 5 of [10]). We also observe that al(S) is obviously contained in We use the notation and theorems of [3]. We consider Inndiag(S): it is also a group of Lie type S, in which the maximal tori T have order if T = T n S and d Then d divides q2 -1 and, if A2 (4 ), then As T is abelian, it is therefore clear that if 7r (S) for i > 1, we have t --- s for s E n(q2 - 1). A2 ( 4 ) and G contains an element of Inndiag (S) BS, then by the above argument we can conclude that T( G ) is connected. We can now consider G such that Inndiag (S) n G = S. Let a be in G B S, then a does not belong to Inndiag (,S ); we denote by n a the set of primes dividing We also recall from paragraph 9 of [3] that peJla for any and (if A2 ( 4 ) ).